Exploring the No-Show Paradox for Condorcet Extensions UsingEhrhart Theory and Computer Simulations

Felix BrandtTechnische Universität München

München, Germanybrandtf@in.tum.de

Johannes HofbauerTechnische Universität München

München, Germanyhofbauej@in.tum.de

Martin StrobelNational University of Singapore

Singaporemstrobel@comp.nus.edu.sg

ABSTRACT

Results from voting theory are increasingly used when dealing withcollective decision making in computational multiagent systems.An important and surprising phenomenon in voting theory is theNo-Show Paradox (NSP), which occurs if a voter is better off byabstaining from an election. While it is known that certain votingrules suffer from this paradox in principle, the extent to which it isof practical concern is not well understood.We aim at filling this gapby analyzing the likelihood of the NSP for six Condorcet extensions(Black’s rule, Baldwin’s rule, Nanson’s rule, MaxiMin, Tideman’srule, and Copeland’s rule) under various preference models usingEhrhart theory as well as extensive computer simulations. We findthat, for few alternatives, the probability of the NSP is rather small(less than 4% for four alternatives and all considered preferencemodels, except for Copeland’s rule). As the number of alternativesincreases, the NSP becomes much more likely and which rule ismost susceptible to abstention strongly depends on the underlyingdistribution of preferences.

KEYWORDS

Social choice theory; voting; no-show paradox; Ehrhart theoryACM Reference Format:

Felix Brandt, Johannes Hofbauer, and Martin Strobel. 2019. Exploring theNo-Show Paradox for Condorcet Extensions Using Ehrhart Theory andComputer Simulations. In Proc. of the 18th International Conference on Au-tonomous Agents and Multiagent Systems (AAMAS 2019), Montreal, Canada,May 13–17, 2019, IFAAMAS, 9 pages.

1 INTRODUCTION

Results from voting theory are increasingly used when dealing withcollective decision making in computational multiagent systems[see, e.g. 12, 25, 54]. A large part of the voting literature studies para-doxes in which seemingly mild properties are violated by commonvoting rules. Moreover, there are a number of sweeping impossibili-ties, which entail that there exists no optimal voting rule that avoidsall paradoxes. It is therefore important to evaluate and comparehow severe these paradoxes are in real-world settings. In this paper,we employ sophisticated analytical and experimental methods toassess the frequency of the No-Show Paradox (NSP), which occursif a voter is better off by abstaining from an election [28]. The ques-tion we address goes back to Fishburn and Brams [28], who writethat “although probabilities of paradoxes have been estimated inother settings, we know of no attempts to assess the likelihoods of

Proc. of the 18th International Conference on Autonomous Agents and Multiagent Systems(AAMAS 2019), N. Agmon, M. E. Taylor, E. Elkind, M. Veloso (eds.), May 13–17, 2019,Montreal, Canada. © 2019 International Foundation for Autonomous Agents andMultiagent Systems (www.ifaamas.org). All rights reserved.

the paradoxes of preferential voting discussed above, and wouldpropose this as an interesting possibility for investigation. Is itindeed true that serious flaws in preferential voting such as theNo-Show Paradox [. . . ] are sufficiently rare as to cause no prac-tical concern?” It is well-known that all Condorcet extensions, alarge class of attractive voting rules, suffer from the NSP and thisis often used as an argument against Condorcet extensions. Ouranalysis covers six Condorcet extensions: Black’s rule, Baldwin’srule, Nanson’s rule, MaxiMin, Tideman’s rule, and Copeland’s rule.

In principle, quantitative results on voting paradoxes can be ob-tained via three different approaches. The analytical approach usestheoretical models to quantify paradoxes based on certain assump-tions about the voters’ preferences such as the impartial anonymousculture (IAC)model, in which every anonymous preference profile isequally likely. Analytical results usually tend to be quite hard to ob-tain and are limited to simple—and often unrealistic—assumptions.The experimental approach uses computer simulations based onunderlying stochastic models of how the preference profiles aredistributed. Experimental results have less general validity thananalytical results, but can be obtained for arbitrary distributions ofpreferences. Finally, the empirical approach is based on evaluatingreal-world data to analyze how frequently paradoxes actually occuror how frequently they would have occurred if certain voting ruleshad been used for the given preferences. Unfortunately, only verylimited real-world data for elections is available.

We analytically study the NSP under the assumption of IACvia Ehrhart theory, which goes back to the French mathematicianEugène Ehrhart [24]. The idea of Ehrhart theory is to model thespace of all preference profiles as a discrete simplex and then countthe number of integer points inside of the polytope defined by theparadox in question. The number of these integer points can bedescribed by so-called quasi- or Ehrhart-polynomials, which canbe computed with the help of computers. The computation of thequasi-polynomials that arise in our context is computationally verydemanding, because the dimension of the polytopes grows super-exponentially in the number of alternatives and was only madepossible by recent advances of the computer algebra system Nor-maliz [16]. We complement these results by extremely elaboratesimulations using four common preference models in addition toIAC (IC, urn, spatial, and Mallows). In contrast to existing results,our analysis goes well beyond three alternatives.

2 RELATEDWORK

TheNSPwas first observed by Fishburn and Brams [28] for a votingrule called single-transferrable vote (STV). Moulin [44] later provedthat all Condorcet extensions are prone to the NSP; the correspond-ing bound on the number of voters was recently tightened by Brandt

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et al. [13]. Similar results were obtained for weak preferences andstronger versions of the paradox [23, 48]. The NSP was also trans-ferred to other settings including set-valued voting rules [see, e.g.,8, 33, 49, 50], probabilistic voting rules [see, e.g., 1, 4, 9, 11, 32] andrandom assignment rules [2, 10].1

The frequency of the NSP was first studied by Ray [52], who, inline with Fishburn and Brams’s classic paper, analyzed situationswhere STV can be manipulated in elections with three alternatives.A similar goal was pursued by Lepelley and Merlin [39] who quan-tified occurrences of the NSP assuming preferences are distributedaccording to IC or IAC. However, in contrast to the present paper,Lepelley and Merlin employed different statistical techniques toestimate the likelihood of multiple variants of the paradox andfocused on score-based runoff rules in elections with three alterna-tives. In a recent paper, this setting was revisited by Kamwa et al.[34] who focused on single-peaked preferences, where alternativescan be ordered on a one-dimensional axis and voters’ preferencesare determined by proximity to their optimal point on this axis.Under this assumption, they found that multiple scoring runoffrules do not suffer from any variant of the NSP while for others,e.g., plurality runoff, the probabilities of a paradox to occur aresignificantly lower compared to the unrestricted domain.

The general idea to quantify voting paradoxes via IAC has beenaround since the formal introduction of this preference model byGehrlein and Fishburn [29] [see, e.g., 36, 37, 40]. Still, it took a good30 years until the connection to Ehrhart theory [24] was establishedby Lepelley et al. [38]. We refer to Gehrlein and Lepelley [30] for amore profound explanation of all details and an overview of resultssubsequently achieved [see also, e.g., 36, 55, 59]. The step from threeto four alternatives, i.e., from six to 24 dimensions, was only madepossible through recent advances in computer algebra systems byDe Loera et al. [21] and Bruns and Söger [18]. Brandt et al. [14] useda framework similar to ours to study the frequency of two single-profile paradoxes (the Condorcet Loser Paradox and the AgendaContraction Paradox). In a recent paper, Bruns et al. [17] also madeuse of the possibility to analyze situations with four alternativesand looked at the Condorcet efficiency of plurality and pluralitywith runoff as well as the structure of majority graphs and varyingBorda paradoxes.

Plassmann and Tideman [51] conducted computer simulationsfor various voting rules and paradoxes under a modified spatialmodel in the three-alternative case. To the best of our knowledge,this is—apart from Brandt et al. [14] and Bruns et al. [17]—the onlystudy of Condorcet extensions from a quantitative angle.

3 PRELIMINARIES

Let A be a set ofm alternatives and N = {1, . . . ,n} a set of voters.We assume that every agent i ∈ N is endowed with a preferencerelation ≻i over the alternatives A. More formally, ≻i is a complete,asymmetric and transitive binary relation, ≻i ∈ A ×A, which givesa strict ranking over the alternatives. If x ≻i y, we say that i prefersx to y.

A preference profile ≻ is a tuple consisting of one preferencerelation per voter, i.e., ≻ = (≻1, . . . , ≻n ). By ≻−i we denote the

1Interestingly, when considering set-valued or probabilistic voting rules, there areCondorcet extensions immune to the NSP under suitable assumptions [8, 11].

preference profile resulting of voter i abstaining the election,≻−i = (≻1, . . . , ≻i−1, ≻i+1, . . . , ≻n ). Preference profiles are definedas vectors of preference relations. For most purposes, however, theordering within this vector is irrelevant and one can alternativelyconsider multisets of preference relations, so-called anonymouspreference profiles.

For two alternatives x,y ∈ A and a preference profile ≻we definethe majority margin дxy (≻) as

дxy (≻) = |{i ∈ N : x ≻i y}| − |{i ∈ N : y ≻i x}|.

Whenever ≻ is clear from the context we only write дxy . A votingrule is a function f mapping a preference profile ≻ to a singlealternative, f (≻) ∈ A.

Condorcet Extensions. Alternative x ∈ A is a Condorcet winnerif it beats all other alternatives in pairwise majority comparisons,i.e., дxy > 0 for all y ∈ A \ {x}. Similarly, x is a weak Condorcetwinner if it beats or ties all other alternatives, i.e., дxy ≥ 0 for ally ∈ A \ {x}. If a voting rule always selects the Condorcet winnerwhenever one exists, it is called a Condorcet extension. A weakCondorcet extension returns a weak Condorcet winner whenever(at least) one exists. Clearly, every weak Condorcet extension isa Condorcet extension. A wide variety of Condorcet extensionshas been studied in the literature [see, e.g., 12, 27]. In this paper,we consider six Condorcet extensions: Black’s rule, Baldwin’s rule,Nanson’s rule, MaxiMin, Tideman’s rule, and Copeland’s rule. Themain criteria for selecting these rules were discriminability (in orderto minimize the influence of lexicographic tie-breaking), simplicity(to allow for Ehrhart analysis and because voters generally prefer‘simpler’ rules), and efficient computability (to enable rigorous andcomprehensive simulations).2 In the following, we briefly definethe rules.

Black’s rule [7] selects the Condorcet winner whenever one existsand otherwise returns a winner according to Borda’s rule (Borda’srule itself is no Condorcet extension).

fBlack(≻) ∈

{x if x is a Condorcet winner in ≻

argmaxx ∈A∑y∈A\{x } дxy otherwise.

Baldwin’s rule [5] proceeds in multiple rounds. In each round, wedrop all alternatives with the lowest Borda score and then continuewith the reduced preference profile, which is used to calculateupdated scores. If multiple—but not all—alternatives are tied last,we delete all of them. Baldwin’s rule chooses one of the alternativesremaining when no more alternative can be removed.

Nanson’s rule [45, 46] is similar to Baldwin’s rule in so far as italso focuses on the Borda scores and gradually eliminates alterna-tives. However, in contrast to before, we now remove all alternativeswith average or below-average Borda score in every round. Nan-son’s rule returns an alternative out of those remaining when allalternatives have identical score.

The MaxiMin rule [7], which is also known as the Simpson-Kramer method [35, 56], looks at the worst pairwise majority com-parison for each alternative. It then returns an alternative withmaximal such score, formally

fMaxiMin(≻) ∈ argmaxx ∈Aminy∈A\{x } дxy .2Note that other discriminating Condorcet extensions such as Kemeny’s rule, Dodg-son’s rule, and Young’s rule are NP-hard to compute [see, e.g., 12].

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Tideman’s rule [57] focuses on the sum of all pairwise majoritydefeats. It yields an alternative where this sum is closest to zero interms of absolute value, i.e.,

fTideman(≻) ∈ argmaxx ∈A∑y∈A\{x } min(0,дxy ).3

Copeland’s rule [19] only relies on the majority relation and notthe exact majority margins. It chooses an alternative where thenumber of majority wins plus half the number of majority draws ismaximal:

fCopeland(≻) ∈ argmaxx ∈A

|{y ∈ A : дxy > 0}|+ 1/2 |{y ∈ A : дxy = 0}|

In order to obtain well-defined voting rules we employ alphabetictie-breaking for all rules defined above. Note that the alphabeticordering does not influence our results as long as we assume thatthere is some underlying ordering. This changes if we allow for tie-breaking based on the preference profile or choice set, somethingwehowever want our tie-breaking to be independent of. All presentedvoting rules can be computed in polynomial time and do not relyon the exact preference profile ≻ but only on the majority marginsthat can conveniently be represented by a skew-symmetric matrixor a weighted directed graph.

Strategic Abstention. A voting rule f is manipulable by strate-gic abstention if there exist some N , A, and ≻ such that for somei ∈ N , f (≻−i ) ≻i f (≻). Given an occurrence of manipulability bystrategic abstention, f is said to suffer from the No-Show Paradox(NSP) (for N , A, ≻). Slightly abusing notation, we also say that ≻is prone to the NSP whenever f , N , and A are clear from the con-text. All rules defined here are Condorcet extensions and thereforemanipulable by strategic abstention. Occurrences of the NSP forBlack’s, Baldwin’s, and Copeland’s rule require three alternativeswhile four alternatives are needed for MaxiMin as well as Nanson’sand Tideman’s rule.

It is interesting to note that whenever a Condorcet winner exists,no weak Condorcet extension allows for manipulation by strategicabstention by a single voter. To see this, assume alternative x is theCondorcet winner, i.e., x wins in a pairwise majority comparisonagainst all other alternatives. While some of these strict majoritypreferences might turn to indifferences if voter i abstains fromthe election procedure, this can only happen for comparisons toalternatives less preferred than x according to ≻i . Hence, everyalternative strictly more preferred than x still loses at least thepairwise majority comparison against x , which remains a weakCondorcet winner. We deduce that irrespective of other possibleweak Condorcet winners and the underlying tie-breaking, no al-ternative preferred to x can be chosen. Of the rules defined above,MaxiMin and Tideman’s rule are weak Condorcet extensions.4

3Tideman’s rule is arguably the least well-known voting rule presented here. It wasproposed to efficiently approximate Dodgson’s rule and is not to be confused withranked pairs which is sometimes also called Tideman’s rule. Also note that the ‘dual’rule returning alternatives for which the sum of weighted pairwise majority wins ismaximal is not a Condorcet extension.4For both MaxiMin and Tideman’s rule this holds by the observation that a weakCondorcet winner does not lose any pairwise majority comparison. Black’s rule failsto be a weak Condorcet extension by definition; a counterexample for Baldwin’s,Nanson’s, and Copeland’s rule is given by Fishburn [27].

Stochastic Preference Models. When analyzing properties of vot-ing rules, it is a common approach to sample preferences accordingto some underlying model. Various concepts to model preferenceshave been introduced over the years; we refer to, e.g., Critchlow et al.[20] and Marden [42] for a detailed discussion. We focus on threeparameter-free models, impartial culture (IC) where each voter’spreferences are drawn uniformly at random, impartial anonymousculture (IAC) [29] where anonymous preference profiles are drawnuniformly at random, and the two-dimensional spatial model wherewe uniformly sample points in the unit square and their proximitydetermines the voters’ preferences. Furthermore, we consider theurn model [6] with parameter 10 and Mallows’ model [41] withϕ = 0.8.

The preference models we consider (such as IC, IAC, and theMallows model) have also found widespread acceptance for theexperimental analysis of voting rules within the multiagent systemsand AI community [see, e.g., 3, 14, 15, 31, 47].

4 QUANTIFYING THE NO-SHOW PARADOX

The goal in this paper is to quantify the frequency of the NSP, i.e., toinvestigate for how many preference profiles a voter is incentivizedto abstain from an election. In order to achieve this goal, we employan exact analysis via Ehrhart Theory and experimental analysis viasampled preference profiles.

4.1 Exact Analysis via Ehrhart Theory

The imminent strength of exact analysis is that it gives reliabletheoretical results. On the downside, precise computation is onlyfeasible for very simple preference models and for small values ofm. We focus on IAC and make use of Ehrhart theory.

First, note that an anonymous preference profile is completelyspecified by the number of voters sharing each of them! possiblerankings onm alternatives. Hence, we can uniquely represent ananonymous profile by an integer point x in a space ofm! dimen-sions. We interpret xi as the number of voters who share rankingi , where rankings are ordered lexicographically. For fixedm, ourgoal is to describe all profiles that are prone to the NSP by usinglinear (in)equalities that describe a polytope Pn .5 Given that this ispossible, the fraction of profiles prone to the NSP can be computedby dividing the number of integer points contained in Pn by thetotal number of profiles for n voters, i.e., the number of integerpoints x satisfying xi ≥ 0 for all 1 ≤ i ≤ m! and

∑1≤i≤m! xi = n.

While the latter number is known to be(m!+n−1m!−1

), the former

can be determined using Ehrhart theory. Ehrhart [24] shows thatit can be found by so-called Ehrhart- or quasi-polynomials f —acollection of q polynomials fi of degree d such that f (n) = fi (n) ifn ≡ i mod q. Obtaining f is possible via computer programs likeLattE [22] orNormaliz [16]. Brandt et al. [14] give a more detaileddescription of the general methodology.

In order to illustrate this method first consider Copeland’s rulein elections with three alternatives under IAC. For the modelingwe need to give linear constraints in terms of voter types—or equiv-alently majority margins—that describe polytopes containing allprofiles prone to the NSP.

5More precisely, Pn is a dilated polytope depending on n, Pn = nP = {n ®x : ®x ∈ P }.

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We first distinguish between the six possible manipulations fromx toy, x , y ∈ A = {a,b, c}. A case-by-case analysis shows that outof these, only manipulations from a to b or c and from either b orc to a are possible. In particular, for each of these cases, there isexactly one voting situation admitting an occurrence of theNSP. Wefind that we can specify the respective profiles using one polytopeeach:

дba ≥ 2, дac ≥ 1, дcb = 1, x6 ≥ 1 (P1)

дca ≥ 2, дab ≥ 1, дbc = 1, x4 ≥ 1 (P2)

дac ≥ 2, дba ≥ 1, дbc = 0, x1 ≥ 1 (P3)

дab ≥ 1, дca ≥ 1, дbc = 0, x2 ≥ 1 (P4)

For the sake of readability we here omit but implicitly assume thatthe total number of voters present is n and there is a nonnegativeamount of voters per voter type. Note that P1 and P2 require nto be odd while every profile contained in P3 or P4 contains aneven number of voters. The total number of anonymous preferenceprofiles admitting a manipulation by abstention is given by thenumber of integer points contained in polytopes P1 to P4.

When considering different rules or a larger amount of alterna-tives, we find that the number of polytopes as well as the numberof linear constraints defining them grows rapidly. Black’s rule, forinstance, can only be manipulated from a Condorcet winner to aBorda winner or vice versa. This distinction is also one of voterparity: a manipulation away from a Condorcet winner is possiblefor odd n, while n is required to be even in the converse case. Incontrast to before, Black’s rule allows for a manipulation betweenany pair of alternatives regardless of n. Hence, we obtain a total of12 polytopes, one for every possible manipulation and parity of n.We only depict the polytope describing manipulations from a to bfor even n in order to give an impression of the general form andomit the rest due to space constraints:

дab + дac ≥ дba + дbc , дba ≥ 1, x6 ≥ 1,дab + дac ≥ дca + дcb , дbc = 0

Here, the inequalities in the left columnmodel that a currently is theBorda winner. The (in)equalities in the second column guaranteethat a manipulator can make b Condorcet winner by abstaining aswell as that with him being present, there is no Condorcet winner.The last column only demands presence of a voter of type ≻6, theonly type able to manipulate.

When moving to MaxiMin and four alternatives, determining thenecessary polytopes becomes tedious. Since alphabetic tie-breakingrules out most symmetries, we need 168 disjoint polytopes of vary-ing sizes to encompass all profiles prone to the NSP. Each of theseis defined by 8 to 10 constraints, not counting the total number ofvoters and nonnegative amount per type.

This approach of modeling profiles prone to the NSP is substan-tially more involved than using Ehrhart theory for other paradoxes,e.g., the Condorcet Loser Paradox [14], because of three reasons.

(i) An occurrence of the NSP requires the presence of a certaintype of voter.

(ii) Preference profiles for which different types of voters areable to manipulate must be counted only once.6

6This effect is only relevant when there are at least four alternatives.

(iii) Possible manipulations not only rely on the winning alter-native itself but on all majority margins that have to adhereto different constraints.

4.2 Experimental Analysis

In contrast to exact analysis, the experimental approach relies onsimulations to grasp the development of different phenomena undervarying conditions. On the upside, this usually allows for resultsfor more complex problems or a larger scale of parameters, bothof which might be prohibitive for exact calculations. At the sametime, however, we find that we need a huge number of simulationsper setting to get sound estimates which in turn often requires ahigh-performance computer and a lot of time. Also, there remainsthe risk that even a vast amount of simulations fails to capture onespecific, possibly crucial, effect.

Regarding the pivotal question of our paper, the frequency ofthe NSP for various voting rules, we sample preference profilesfor different combinations of n andm using the modeling assump-tions explained in Section 3. Our simulations were conducted ona XeonE5-2697 v3 with 2 GB memory per job. The total runtimeeasily accumulates to thirty years on a single-processor machine.

5 RESULTS AND DISCUSSION

In this sectionwe present our results obtained by both exact analysisand computer simulations.

5.1 Analytical Results under IAC

We first focus on Copeland’s rule with three alternatives, as ourmodeling in Section 4.1 allows for an exact analysis of the NSP.In particular, we compute the following Ehrhart-polynomial f (n)with period q = 2:

f0(n) = 1/192n4 − 1/48n3 − 1/48n2 + 1/12n

f1(n) = 1/192n4 − 5/96n2 + 3/64

Recall that f (n) = fi (n) if n ≡ i mod q. Consequently, the fractionof profiles that admit a manipulation by strategic abstention is givenby

f0(n)(n+55 )

if n is even and f1(n)(n+55 )

if n is odd.

This frequency of the NSP for Copeland’s rule andm = 3 is plottedin Figure 1, together with results obtained by computer simulations.

With respect to Black’s rule andm = 3, we obtain an Ehrhart-polynomial with slightly larger period q = 6. Once more, we canexplicitly give f (n) which looks as follows:

f0(n) = 1/192n4 − 5/48n2

f1(n) = 1/192n4 − 1/48n3 − 7/96n2 + 3/16n − 19/192

f2(n) = 1/192n4 − 5/48n2 + 1/3

f3(n) = 1/192n4 − 1/48n3 − 7/96n2 + 3/16n + 15/64

f4(n) = 1/192n4 − 5/48n2 + 1/3

f5(n) = 1/192n4 − 1/48n3 − 7/96n2 + 3/16n + 15/64

The fraction of profiles prone to the NSP for Black’s rule andm = 3 is visualized in the middle part of Figure 1.

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100 101 102 1030%

0.5%

1%

1.5%

2%

(13, 1.63%)

Copeland’s rule (m = 3)

Polynomial

Simulation

100 101 102 1030%

0.5%

1%

1.5%

2%

(14, 1.55%) (16, 1.55%)

Black’s rule (m = 3)

Polynomial

Simulation

100 101 102 1030%

0.2%

0.4%

0.6% (14, 0.55%)

MaxiMin (m = 4)

Polynomial

Simulation

Figure 1: Profiles prone to the NSP for Copeland’s rule,

Black’s rule, and MaxiMin; fixedm, increasing n

Similar connections between analytical and experimental resultsfor MaxiMin can be observed in the bottom part of Figure 1. Notethat while we are able to explicitly give the Ehrhart-polynomials forCopeland’s and Black’s rule andm = 3 here, this is not possible forMaxiMin andm = 4 due to space constraints. The correspondingpolynomial f (n) has a period of q = 55 440, i.e., it consists of 55 440different polynomials.We deduce that no two points in theMaxiMinchart of Figure 1 are computed via the same polynomial, whichmakes the regularity of the curve even more remarkable.

A couple of points come to mind when closely studying thesegraphs. First, we note that the results obtained by simulation almostperfectly match the exact calculations, which can be seen as strongevidence for the correctness of both. On the one hand, it confirmsour modeling via polytopes, and at the same time highlights thatwe are running a sufficiently large amount of simulations. Whilethis does not bear definite testimony to the correctness for largerm,we highlight that our implementation is both generic (with respectto m and n) and not particularly complex, which minimizes therisk of errors. We additionally believe that the perfect smoothnessof Figure 2 together with the fact that the NSP is independent ofn, m, and the underlying voting rule strongly suggests that ourexperimental results are sound and reliable.

We see that for Black’s rule the maximum is attained at 14 and16 voters with 1.55% of all profiles suffering from the NSP. ForCopeland’s rule themaximum is at 13 voters and 1.63% of all profiles,while forMaxiMin andm = 4 it is at 14 voters and a fraction of 0.55%of profiles. Hence, we can argue that for elections with very fewalternatives, the NSP seems to hardly cause a problem, independentof the number of voters or the voting rule considered. Strikingly,the maxima occur at roughly the same number of voters, with thisnumber varying between being even or odd. Also observe thatBlack’s and Copeland’s rule are more sensitive to the parity of nthan MaxiMin.

Furthermore, we note that the probability for the NSP to occurconverges to zero as n goes to∞; this holds true for all voting rulesconsidered and all fixedm. Intuitively, this is to be expected as forlarger electorates, a single voter’s power to sway the result dimin-ishes. This first idea can be confirmed by considering the respectivemodeling via polytopes. Each modeling will contain at least oneequality constraint, e.g., in the third column of our modeling ofCopeland’s rule in Section 4.1. Consequently, the polytopes describ-ing profiles for which a manipulation is possible are of dimensionat most m! − 1. By Ehrhart [24], this means that the number ofthose profiles can be described by a polynomial of n of degree atmostm! − 1. The total number of profiles, on the other hand, canequivalently be determined via a polynomial of degreem!. Hence,the fraction of profiles prone to theNSP is upper-bounded byO(1/n).Following the intuitive argument, similar behavior is to be expectedfor all reasonable preference models and voting rules.

Form = 4, determining the Ehrhart polynomials for both Black’sas well as Tideman’s rule proved to be infeasible, even when us-ing a custom-tailored version of Normaliz and employing a high-performance cluster.7 Copeland’s rule unfortunately causes prob-lems even earlier: for four alternatives the modeling via linear

7For Black’s rule, we find that the polynomial would be of period q ≈ 2.7 · 107corresponding to a mid two-digit GB file size.

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(in)equalities quickly becomes very challenging due to the ruleonly caring about unweighted majority comparisons. For all rules,m ≥ 5 appears to be out of scope for years to come.

5.2 Experimental Results under IAC

In this section, we rely on simulations to grasp how often theNSP can occur for different combinations of n andm up to 50 votersand 30 alternatives. Our results can be found in Figure 2 and allowfor the following observations to be made.

To begin with, the relatively low fraction of profiles prone tothe NSP for Copeland’s rule, Black’s rule, and MaxiMin with asmall number of alternatives increases asm grows. This increase isquite dramatic for Copeland’s rule and MaxiMin. In particular, foronly 20 alternatives and both rules, a rough quarter of all profilesadmit manipulation by abstention for a medium count of voters.This number is too large to discard the NSP as merely a theoreti-cal problem. Black’s rule, on the other hand, remains stable on acomparatively moderate level. Felsenthal and Nurmi [26] argue infavor of Nanson’s rule as it is—in contrast to the related Baldwin’srule—not prone to the NSP for three alternatives. We show that thisdifference between the two rules becomes moot for larger numbersof alternatives: the fractions of profiles allowing for a manipulationare on a roughly identical, severely high level.8 This shows thatvoting rules based on Borda scores do not necessarily fare betterwith respect to the NSP.

When examining Baldwin’s rule in Figure 2, the ridge at n = 3immediately catches the observer’s eye.We conjecture this unique behavior of Baldwin’srule is due to preference profiles similar in struc-ture to the one depicted on the right. In case voter3 places sufficiently many alternatives over x , x isgoing to be eliminated on the way causing y to even-tually be chosen. Then again, if voter 3 abstains, xis always going to be selected as long as it beats yin the tie-breaking order. Note that x and y can bechosen almost freely, all other alternatives placedvirtually arbitrarily, and many profiles only similarin structure also work.

1 2 3

x y...

y x.........

y

x...

Especially when considering Black’s, Tideman’s, and Copeland’srule, we see that the parity of n crucially influences the results.However, the parity of n does not affect the fractions in a consistentway: higher fractions occur for Black’s and Copeland’s rule when nis even, in contrast to Tideman’s rule where this happens when n isodd. For Black’s rule, this is most probably due to the fact that therearemore suitable profiles close to having a Condorcet winner (дxy =0) than profiles close to not having one (дxy = 1).9 Copeland score’sare integers when the number of voters is odd and half-integerswhen the number of voters is even. Hence, differences betweenalternatives are potentially more distinct for an odd number ofvoters which we assume makes manipulations harder to achieve.For Tideman’s rule, we currently lack a convincing explanation for

8Felsenthal and Nurmi [26] also show that none of the two rules fares strictly betterthan the other. Indeed, there are profiles where a manipulation is possible accordingto Baldwin’s rule but not using Nanson’s rule and vice versa.9For Black’s rule, manipulation is only possible either towards or away from a Con-dorcet winner since Borda’s rule is immune to strategic abstention and manipulationis impossible from Condorcet winner to Condorcet winner.

the observed behavior, mostly because it is hard to intuitively graspwhen exactly a preference profile is manipulable.

Regarding Baldwin’s and Nanson’s rule as well as MaxiMin,the parity of n seems to have little effect on the numbers. Moredetailed analysis shows that at least for MaxiMin this appearance isdeceptive: when manipulating towards an alphabetically preferredalternative, fractions are higher for even n, while the contraryholds for manipulations towards an alphabetically less preferredalternative. In sum, these two effects approximately cancel eachother out.

The flawless smoothness and regularity of all plots in Figure 2are due to 106 runs per data point. This large number allows for all95% confidence intervals to be smaller than 0.2%. Our simulationstook 35 to 48 hours for each data point and there are 1 500 datapoints per plot.

5.3 Comparing Different Preference Models

In order to get an impression of the frequency of the NSP underdifferent preference models we fix the number of alternatives tobem = 4 orm = 30 and sample 106 profiles for increasing n up to1 000 or 200, respectively.10 Figure 3 gives the fraction of profilesprone to the NSP using either MaxiMin, Black’s, or Tideman’s rule.

A close inspection of these graphs allows for multiple conclu-sions. First, we see that in particular Black’s rule shows a severedependency on the parity of n. For better illustration, we depicttwo lines per preference model to highlight this effect; which linestands for odd and which for even n is easiest checked using theircorresponding point of intersection with the x-axis, which is either1, 2, or 3 throughout. Apart from explanations given earlier, it isnot completely clear why differences are more prominent for somevoting rules, why we sometimes see higher percentages for oddn and other times for even n, or why for some instances there isa large discrepancy for one preference model but hardly any foranother.

IC and IAC are often criticized for being unrealistic and onlygiving worst-case estimates [see, e.g., 53, 58]. This criticism is gen-erally confirmed by our experiments, which show that the highestfractions of profiles is prone to the NSP when the sampling is doneaccording to IC or IAC. A notable exception is Black’s rule for 30alternatives, where a different effect prevails: for many alternativesand comparably few voters, situations in which a Condorcet win-ner (almost) exists appear less frequently under IC or IAC thanunder the other preference models. In absence thereof, Black’s rulecollapses to Borda’s rule, which is immune to the NSP. Note thatwere we to conduct a dual experiment with fixed n and increasingm, the fraction of profiles prone to the NSP using Black’s rule andIC or IAC would converge to zero for similar reasons.

Wemoreover see that IC, IAC, and the urnmodel exhibit identicalbehavior for m = 30. The second and fourth column of Figure3 therefore seem to only feature three preference models, eventhough all five are depicted. This may be surprising at first but isto be expected since IC and IAC can equivalently be seen as urnmodels with parameters 0 and 1, respectively. For 30! ≈ 2.7 · 1032

10For increasingm the computations quickly become very demanding. The valuesform = 30 and n ≥ 99 are determined with 50 000 runs each only. The size of all 95%confidence intervals is, however, still within 0.5%.

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Figure 2: Fraction of profiles prone to the NSP for different rules and increasing n andm

voter types and a comparatively small n the difference betweenparameters 0, 1, and 10 is simply too small for a visible difference.

The large conceptual similarities between Baldwin’s and Nan-son’s rule are also reflected in the corresponding charts. Apart fromthe peak at n = 3 for Baldwin’s rule, both look almost identicalfor all preference models with the small difference being that Nan-son’s rule appears to feature a slightly lower manipulability. Fewerrounds for winner determination thus do not seem to come at acost with respect to the NSP.

Finally, Copeland’s, Baldwin’s, and Nanson’s rule as well as Maxi-Min to a lesser extent appear to fare exceptionally bad with respectto the NSP and IC, IAC, and the urn model. At the same time, noneof these rules exhibits overly conspicuous behavior for the spatialand Mallows’ model. This suggests that the risk of a possible ma-nipulation is reduced by structural similarities in the individualpreferences compared to a greater likelihood for very diverse rank-ings. Though generally in line with expectations, we currently donot have a profound explanation for the magnitude of this effect.For Copeland’s rule, it is plausible to assume that its particularlybad performance results from the rule using less information, i.e.,Copeland’s rule is the only majoritarian rule considered here.

The maximal fraction of total profiles prone to theNSP form = 4,m = 30, different voting rules, preference models, and varying val-ues of n is given in Table 1. Among other things, we for instancenote that the maxima constantly occur for a higher number of vot-ers for IC (26 to 51 voters) than for Mallows’ model (3 to 17 voters),a fact probably due to an increasing (expected) structure underMallows’ model and larger n.

m IC IAC Spatial Urn Mallows

Black 4 3.92(29) 3.73(18) 1.62(15) 2.30(10) 3.23(17)

30 5.12(22) 5.12(22) 7.70(17) 5.14(20) 9.90(13)

Baldwin 4 3.92(27) 3.07(23) 0.40(15) 0.84(12) 2.14(13)

30 35.4(49) 35.4(51) 2.54(21) 35.7(49) 5.73(3)

Nanson 4 3.64(27) 2.76(24) 0.44(13) 0.68(16) 2.20(14)

30 34.9(51) 34.8(51) 2.38(21) 34.7(99) 3.40(12)

MaxiMin 4 1.00(30) 0.56(14) 0.14(3) 0.13(3) 0.50(10)

30 28.0(30) 28.0(30) 2.31(3) 28.0(30) 3.01(6)

Tideman 4 0.80(26) 0.67(5) 0.19(5) 0.32(5) 0.62(3)

30 15.6(51) 15.6(49) 2.42(7) 15.6(49) 4.12(3)

Copeland 4 6.96(29) 5.54(20) 0.91(14) 2.07(13) 4.13(16)

30 31.2(50) 31.0(50) 4.28(21) 31.1(50) 6.33(16)

Table 1: Maximal percentage of total profiles prone to the

NSP for different combinations of voting rules and prefer-

ence models with m = 4 or m = 30; the number of voters nfor which the maximum occurs attached in parentheses

5.4 Empirical Analysis

We have also analyzed the NSP for empirical data obtained fromreal-world elections. Unfortunately, such data is generally rela-tively sparse and imprecise and often only fragmentarily available.A check of all 315 strict profiles contained in the PrefLib library[43] for occurrences of the NSP shows that two profiles admit amanipulation by abstention when Black’s rule is used, one profile

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m = 4 Black’s rule m = 30 m = 4 Baldwin’s rule m = 30

100 101 102 1030%

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IC IAC Urn Spatial Mallows

Figure 3: Profiles prone to the NSP for different rules, fixedm, and increasing n on the x-axis; two lines per preference model

depending on the parity of n; IC, IAC and the urn model collapse form = 30, resulting in a bluish grey line

for each Copeland’s, Baldwin’s, and Nanson’s rule, and that nomanipulation is possible for MaxiMin as well as Tideman’s rule.11While this suggests a low susceptibility to the NSP in real-worldelections, much more data would be required to allow for meaning-ful conclusions.

6 CONCLUSION

We analyzed the likelihood of the NSP for six Condorcet extensions(Black’s, Baldwin’s, and Nanson’s rule, MaxiMin, and Tideman’sas well as Copeland’s rule) under various preference models usingEhrhart theory as well as extensive computer simulations and someempirical data. Our main results are as follows.

• When there are few alternatives, the probability of the NSPis almost negligible (whenm = 4, less than 1% for MaxiMinand Tideman’s rule, less than 4% for Black’s, Baldwin’s, andNanson’s rule, and less than 7% for Copeland’s rule underall considered preference models).

• When there are 30 alternatives and preferences are mod-eled using IC, IAC, and the urn model, Black’s rule is least

11For instance the profile allowing for a manipulation under Copeland’s rule is immuneto the NSP for all other rules. It features 10 alternatives and 30 voters. Baldwin’s andNanson’s rule exhibit the NSP for the same profile.

susceptible to the NSP (< 6%), followed by Tideman’s rule(< 16%), MaxiMin (< 29%), Copeland’s rule (< 32%) Nanson’srule (< 35%) , and Baldwin’s rule (< 36%).

• For 30 alternatives and the spatial and Mallows’ model, thisordering is roughly reversed. MaxiMin and Nanson’s rule areleast susceptible (< 4%), followed by Tideman’s rule (< 5%),Baldwin’s rule (< 6%), Copeland’s rule (< 7%), and Black’srule (< 10%).

• The parity of the number of voters significantly influencesthe manipulability of Black’s, Tideman’s, and Copeland’srule. Black’s and Copeland’s rule are more manipulable foran even number of voters whereas MaxiMin is more manip-ulable for an odd number of voters (under the IAC assump-tion).

• Whenever analysis via Ehrhart theory is feasible, the resultsare perfectly aligned with our simulation results, highlight-ing the accuracy of the experimental setup.

• Only four (out of 315) strict preference profiles in the Pref-Lib database are manipulable by strategic abstention (ma-nipulations only occur for Black’s, Baldwin’s, Nanson’s, andCopeland’s rule, but not for MaxiMin and Tideman’s rule).

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